The overwhelmingly vast majority of truth cannot be expressed by language
Human language is countably infinite because:
Now consider the following example of the "subset statement":
This statement is true because 56, 134, 255, and 533 are natural numbers.
Now, consider that there are uncountably infinite subsets of the natural numbers. Therefore, with language being countably infinite, there are uncountably infinite subsets of the natural numbers for which the "subset statement" cannot be expressed by language.
In "True but Unprovable", Yanofsky writes:
Generally, when truth can be expressed by language, this is a rare exception and not the rule.
- its alphabet is finite
- every string in human language is of finite length
Now consider the following example of the "subset statement":
The set {56, 134, 255, 533} is a subset of the natural numbers.
This statement is true because 56, 134, 255, and 533 are natural numbers.
Now, consider that there are uncountably infinite subsets of the natural numbers. Therefore, with language being countably infinite, there are uncountably infinite subsets of the natural numbers for which the "subset statement" cannot be expressed by language.
In "True but Unprovable", Yanofsky writes:
http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf
This brings to light an amazing limitation of the power of language.
The collection of all subsets of natural numbers is uncountably infinite while the set of expressions describing subsets of natural numbers is countably infinite. This means that the vast, vast majority of subsets of natural numbers cannot be expressed by language.
Some true mathematical facts are expressible while the vast, vast majority of mathematical facts are inexpressible.
Generally, when truth can be expressed by language, this is a rare exception and not the rule.
Comments (37)
Is it ok if we just copy-and-paste the replies? Or should we link to them?
This is not right. Perhaps "the vast, vast majority of subsets of natural numbers cannot be expressed by language," but judgments of true or false only apply to propositions. Propositions are linguistic entities - they can all be expressed in language. If it can't be expressed in language, it isn't a proposition and if it isn't a proposition, it can't be true or false.
The following is a legitimate proposition:
The set {6,8,11} is a subset of the natural numbers.
It is true or false.
Quoting T Clark
The following proposition is tautologically true:
Every subset of the natural numbers is a subset of the natural numbers.
The problem is that most individual subsets of the natural numbers cannot be expressed by language. Some can but most cannot.
The ineffable propositions are still true propositions because all of them are true given the tautology mentioned above.
Quoting T Clark
Propositions that can be expressed by language are indeed linguistic entities. The ones that cannot be expressed by language, however, are not. For example, the general case of "Subset X of the natural numbers is a subset of the natural numbers" is true, irrespective of whether X can be expressed by language or not.
There are no propositions that can't be expressed in language.
Quoting Tarskian
This is just a restatement of the tautological proposition "All subsets of the natural numbers are subsets of the natural numbers."
I can see you and I are not going to agree on this. I'll give you the final word.
The distinction between countable and uncountable infinity, originally introduced by Georg Cantor, has always been controversial.
When first confronted with the matter, I do not think that anybody right in his mind agrees on this. It is just too controversial. The first reaction is usually, disgust. It takes quite a while before someone can actually accept this kind of thinking.
Any thoughts on why?
Is it a blow to people's egos to face the limitations of human thought?
In my opinion , it decisively divorces mathematical reality from physical reality, which is otherwise its origin.
Humans, but also animals, have quite a bit of basic arithmetic and logic built into their biological firmware, if only, for reasons of survival. To the extent that mathematics stays sufficiently close to these innate notions, people readily accept its results.
There is no notion of infinity in physical reality. In that sense, Cantor's work is rather unintuitive. You have to learn to think like that. It does not come naturally.
You seem to be forgetting that languages can evolve and it's use can be arbitrary. We can always add more letters to the alphabet and we only communicate what is relevant. Why would we need a word for every natural number if we never end up finding a use for those numbers? If the universe is finite then there is no problem here. If it isn't then the universe at least appears to be consistent in that the physical laws are the same no matter where you go in the universe. Novelty would be the only aspects of the universe needing new terms to describe them.
What is one example of a subset of the natural numbers that cannot be expressed by language?
Also note that mathematical notation is a kind of extension to the natural languages.
Assuming this statement is true, what do you think is its philosophical significance?
:up:
There is a one-to-one mapping between the subsets of the natural numbers and the real numbers. So, we can represent a subset of the natural numbers by its corresponding real number.
We construct the real number as the Ricardian number r:
https://en.m.wikipedia.org/wiki/Richard%27s_paradox
The Ricardian real number r is defined as undefinable and therefore the corresponding subset of the natural numbers cannot be expressed in language either.
If you look at the epistemic JTB account for knowledge as a justified true belief, it means that the overwhelmingly vast majority of true beliefs are ineffable and cannot possibly be justified.
Hence, most truth is not knowledge.
The fact that some truth can be justified is the rare exception and not the rule.
Perhaps your OP topic only indicates that "the epistemic JTB account" is inadequate in some way.
They are ineffable, so they have no opportunity to be beliefs at all, and therefore no occasion to be justified.
Ineffable truths are never believed. And I guess, numerically, most truths are ineffable. But all of these ineffable truths seem quite irrelevant too.
Then there is still the next level: the beliefs about these ineffable beliefs which are not necessarily ineffable. There is a large literature about Richardian numbers even though these numbers are undefinable.
Quoting hypericin
Well, it's a bit like the axiom of infinity, i.e. insisting on the existence of an ineffable cardinality. At first glance, it also looks irrelevant.
Originally, most mathematicians utterly rejected the axiom of infinity and Cantor's work in general:
The ineffable sequence of infinite cardinalities is an essential axiomatic belief in contemporary mathematics, no matter how much it sounds like philosophy or theology.
I don't think that JTB is inadequate.
Most truth cannot be known in terms of JTB. That is not a flaw in JTB. The nature of reality is simply like that.
If we happen to know some truth, then it is the rare exception and not the rule.
You have not shown that this is the case (i.e. a belief that is neither justified nor true).
Oh, my goodness me. How shocking.
Now, can you give an example of one those the truths?
Just one will do. Then we will have an idea of what we are dealing with. Of the import of this startling, enigmatic observation.
Hmm.
Consider the following proposition:
The set X is a subset of the natural numbers.
This is trivially true for an example subset such as {5, 67, 257}.
There are an uncountably infinite number of such subsets. However, there are only a countably infinite number of sentences in language. Therefore, for most subsets X of the natural numbers, this true sentence cannot be expressed in language.
Not on this message board, obviously. But there is a rumour that the mystical can be made manifest. That is what Zen is about, is it not? And the Dao, and the holy.
Talk is cheap and very limited, so one is obliged to wave a hand in the general direction of the uniqueness that is everywhere, all the time.
Construct a Richardian number and map it one-to-one to a subset of the natural numbers. This subset is ineffable:
Quoting Tarskian
Quoting 180 Proof
Heres my unsolicited attempt to answer your question: writing tentatively, with the need for corrective refutation:
Tarskians premise suggests to our understanding that: the mapping from verbal language to experience is categorically incomplete.
In turn, this tells us that sine qua non rules about the volume and thoroughness of verbal databases of information evolve without completion, and that therefore the epistemological project is likewise an evolving project without completion.
One of the important consequences of an epistemological project that never completes is knowledge of truths that cannot be proven. This leads to speculation about the science, math and language databases all being open. If so, it may be the case there is no complete systemization.
If it can be surmised that no systemization of science, math and language can be complete, then it might follow that no correspondence between them can be complete.
In turn, this might suggest the need for a radical overhaul of our definition(s) of truth: if correspondence is always incomplete, then the cognitive vector (thinking about science, math and language) like the physical vector, with its position and momentum coordinates, might be uncertain per Heisenberg.
Every property of this unstateable number is itself an unstateable truth.
Example: Number r is a real number.
If number r is unstateable then this sentence is also unstateable, no matter how true this sentence may be.
I actually took the example of the subsets of the natural numbers literally from Yanofsky:
http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf
Yanofsky argues that the fact that the sentence is ineffable automatically makes it unprovable. This is indeed the case for an individual sentence. The truth of entire set of sentences, however, is provable.
There are truths about sets of sentences that apply to each individual sentence while we do not have the ability to express by language most of such individual sentences.
Im thinking Tarskian is trying to tell you that the inexpressible truth is the paradox of a Ricardian number being simultaneously: a member of the natural numbers/not being a member of the natural numbers, as based upon the unresolvable paradox.
If you claim the paradox itself is the statement of the inexpressible truth, then youre trashing the principle of non-contradiction, and your logical systems crash.
But it isn't.
But it isn't true.
But it isn't true, manifestly.
But it isn't true, manifestly you can go on forever.
But it isn't true manifestly you can go on forever and ever.
And I told him "But it isn't true, manifestly you can go on forever and ever."
We had a discussion, and I told him ""But it isn't true, manifestly you can go on forever and ever."
etc.
How do you know that what you believe in is true if you can't express it?
There's noting novel in the natural numbers not being enumerable. What this shows is that the list from which r is derived cannot be constructed.
What I would like is something that shows these unstatable truths to have some sort of significance. Trouble is, if they have significance (note the word), that significance is statable...
That there are unstatable trivialities is not significant.
"The overwhelmingly vast majority of truth cannot be expressed by language" is ambiguous. Is it to be understood, as I think @Tarskian does, as saying that there are true statements that cannot be stated, (a contradiction), or is it to be understood as that while any particular truth can be stated, not every truth can ever be stated, which is a simple consequence of there being transfinite numbers.
Hence my question - give an example of a truth that cannot be stated. "r is a real" is a truth that can be stated.
I suspect this underpins what was said by and . And sets a puzzle to 's restriction on thought - the paradox of being unable to tell us of something that cannot be said.
Quoting Banno
:smirk:
:roll: Big effin' whup. Nothing new in this insight approximating, not "incompleteness" (another reified / Platonic abstraction) since Eudoxus' method of exhaustion¹ (e.g. squaring the circle). Also, merelogy²: parts (e.g. reason) cannot equal, let alone exceed, the whole (e.g. reality) to which they belong (i.e. in which they are inscribed-entangled) i.e. reality is in our reach yet also exceeds our grasp because we are real and nothing more e.g. Gödel has only axiomatized and Heisenberg / Schödinger have only instrumentalized this formal-merelogical limit that constrains epistemic / cognition (pace Kant).
https://en.m.wikipedia.org/wiki/Method_of_exhaustion [1]
https://en.m.wikipedia.org/wiki/Mereology [2]
Quoting 180 Proof
Approximation can be incomplete, as in the case of pi. More to the point, strategic incompleteness doesnt have a specified boundary it can approach; all correspondences operating under strategic incompleteness are relative without any universal standard of reference, so the field of epistemology as defined by its grammar is such that no one can speak final words about what the attributes of the abstract form should be.
Reification has only a weak form under strategic incompleteness because no systems are finalized into hard boundaries.
Since strategic incompleteness posits only unfinished parts of different sizes incompletely related, with no finalization of systemization:
Quoting 180 Proof
The above doesnt stand as its counter-narrative. The argument that parts cannot exceed their whole is foundational to strategic incompleteness. This limitation is the reason why systemization is incomplete; if not, the part would be able to contain the whole of itself, a paradox. This is why there is no rational origin of anything (and why your Deist god is necessary to initiate existence), and thus the point of view of strategic incompleteness says there is no beginning and no end, only partial approaches to same.
Quoting myself. A bad sign. Might try this with an analogue.
Supose you are building a deck, which will have forty floor boards screwed to joists. You have four hundred floorboards.
Now it's true that the overwhelmingly vast number of floorboards cannot be screwed to joists. But it is not true that any one floorboard cannot be screwed to the joists.
We can see this by asking to be shown a floorboard that cannot be screwed to the joists. And the answer is, they all can.
Similarly, even supposing that it is true that the overwhelmingly vast majority of truths cannot be expressed by language, it does not follow that any particular truth cannot be expressed in language.
So we ask, show an example of a true statement that cannot be stated. And the answer is, they can all be stated.
Though, to split the difference, I agree with
If someone points out, as @Tarskian did, that the set of unexpressed sentences is larger than the set of expressed sentences I'd agree, but would not come to the conclusion that the title of the OP states.
And I wouldn't bother with making statements about "the overwhelmingly vast majority" after that, as obviously those are the words of the bean counters who want a ledger to prove a point, which philosophy doesn't bother with (when it's good).
Well, of course Un's right. @Unenlightened is always right, the bastard. Best just to ignore his posts, else he bring all these threads to an end, leaving us with no alternative but to engage with the real world.
True.
:roll:
Another non sequitur.
Ad hominem. Besides, I'm not a "deist" and do not espouse "deism".
I have neither claimed nor implied that "existence" is/was "initiated".